Matricesthree

Matricesthree is a term used in theories of linear and multilinear algebra to denote a structured three-dimensional array, extending the familiar concept of a matrix to three indices. In this sense, a matricesthree is a third-order tensor whose entries are denoted a_{i j k} with dimensions I × J × K over a field such as the real or complex numbers.

Unlike two-dimensional matrices, matricesthree cannot in general be fully described by a single set of row and column operations. Its frontal, lateral, and horizontal slices form matrices that provide convenient views of the data. Common notation uses mode-n fibers and unfoldings to convert the three-way array into matrices, enabling the application of traditional algebraic techniques. The mode-n product with a matrix on one mode generalizes matrix multiplication to the three-way setting.

Matricesthree are central to multilinear algebra and data analysis, where three-way data arise naturally in time-evolving images, multi-sensor measurements, and relational data with two object indices. Typical tasks include dimensionality reduction via tensor decompositions such as Tucker and CANDECOMP/PARAFAC, tensor completion, and multi-way regression. These methods aim to uncover latent factors that explain variations across all modes while preserving inherent multi-way structure.

Historically, the term matricesthree is used in discussions of higher-order generalizations of matrices and sometimes appears in educational and expository contexts to emphasize the three-dimensional aspect. In practice, the concept is closely aligned with the broader framework of tensors and multiway data analysis, and it complements standard matrix theory rather than replacing it.

Related topics include tensors, three-way arrays, multilinear algebra, tensor decompositions, and tensor contractions.